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Theorem bcthlem2 23931

Description: Lemma for bcth 23935. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

**Hypotheses**
Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
bcthlem.8	⊢ (𝜑 → 𝐶 ∈ 𝑋)
bcthlem.9	⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
bcthlem.10	⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11	⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))

**Assertion**
Ref	Expression
bcthlem2	⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Distinct variable groups: 𝑘,𝑛,𝑟,𝑥,𝑧 𝐶,𝑟,𝑥 𝑔,𝑘,𝑛,𝑟,𝑥,𝑧,𝐷 𝑔,𝐹,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝐽,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝑀,𝑘,𝑛,𝑟,𝑥,𝑧 𝜑,𝑘,𝑛,𝑟,𝑥,𝑧 𝑥,𝑅 𝑔,𝑋,𝑘,𝑛,𝑟,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑔) 𝐶(𝑧,𝑔,𝑘,𝑛) 𝑅(𝑧,𝑔,𝑘,𝑛,𝑟)

**Proof of Theorem bcthlem2**
Step	Hyp	Ref	Expression
1		bcthlem.11	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
2		fvoveq1 7182	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑛 + 1)))
3		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
4		fveq2 6673	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛))
5	3, 4	oveq12d 7177	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘𝐹(𝑔‘𝑘)) = (𝑛𝐹(𝑔‘𝑛)))
6	2, 5	eleq12d 2910	. . . . . 6 ⊢ (𝑘 = 𝑛 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))))
7	6	rspccva 3625	. . . . 5 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
8	1, 7	sylan 582	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
9		bcthlem.9	. . . . . 6 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
10	9	ffvelrnda 6854	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))
11		bcth.2	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
12		bcthlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
13		bcthlem.5	. . . . . . 7 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
14	11, 12, 13	bcthlem1 23930	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
15	14	expr 459	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘𝑛) ∈ (𝑋 × ℝ⁺) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))))
16	10, 15	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
17	8, 16	mpbid 234	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))
18		cmetmet 23892	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
19	12, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
20		metxmet 22947	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
22	11	mopntop 23053	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
24		xp1st 7724	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋)
25		xp2nd 7725	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ⁺)
26	25	rpxrd 12435	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)
27	24, 26	jca 514	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*))
28		blssm 23031	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
29	28	3expb 1116	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
30	21, 27, 29	syl2an 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
31		1st2nd2 7731	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (𝑔‘(𝑛 + 1)) = ⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
32	31	fveq2d 6677	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩))
33		df-ov 7162	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
34	32, 33	syl6reqr 2878	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
35	34	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
36	11	mopnuni 23054	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
37	21, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
38	37	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → 𝑋 = ∪ 𝐽)
39	30, 35, 38	3sstr3d 4016	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽)
40		eqid 2824	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	sscls 21667	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
42	23, 39, 41	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
43		difss2 4113	. . . . . . . 8 ⊢ (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
44		sstr2 3977	. . . . . . . 8 ⊢ (((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
45	42, 43, 44	syl2im 40	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
46	45	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))))
47	46	ex 415	. . . . 5 ⊢ (𝜑 → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))))
48	47	3impd 1344	. . . 4 ⊢ (𝜑 → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
49	48	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
50	17, 49	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
51	50	ralrimiva 3185	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∖ cdif 3936 ⊆ wss 3939 ⟨cop 4576 ∪ cuni 4841 class class class wbr 5069 {copab 5131 × cxp 5556 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 1^st c1st 7690 2^nd c2nd 7691 1c1 10541 + caddc 10543 ℝ^*cxr 10677 < clt 10678 / cdiv 11300 ℕcn 11641 ℝ⁺crp 12392 ∞Metcxmet 20533 Metcmet 20534 ballcbl 20535 MetOpencmopn 20538 Topctop 21504 Clsdccld 21627 clsccl 21629 CMetccmet 23860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-bases 21557 df-cld 21630 df-cls 21632 df-cmet 23863

This theorem is referenced by: bcthlem3 23932 bcthlem4 23933

W3C validator